In the radio frequency (RF) engineering field, the impedance transformation effect results from the connection of a transmission line to a mismatched load impedance. However, it is commonly assumed that the impedance transformation effect is negligible when the transmission line length is small, meaning that the transmission line length is much smaller than a particular wavelength of excitation.
Turning now to FIG. 1, a diagram of a partial circuit is shown and is generally identified by reference numeral 8. The circuit 8 has an input impedance Zin, a load impedance ZL, and a transmission line with a characteristic impedance Z0, a phase constant β and a physical length Δz. The circuit 8 is driven by a sinusoidal power signal from a power source at a particular wavelength of excitation. The transmission line is assumed to be effectively lossless. The relationship between the input impedance Zin and the load impedance ZL is expressed as:
                              Z          in                =                              Z            0                    ⁢                                                    Z                L                            +                                                jZ                  0                                ⁢                                  tan                  ⁡                                      (                                          βΔ                      ⁢                                                                                          ⁢                      z                                        )                                                                                                      Z                0                            +                                                jZ                  L                                ⁢                                                                  ⁢                                  tan                  ⁡                                      (                                          βΔ                      ⁢                                                                                          ⁢                      z                                        )                                                                                                          (                  Equation          ⁢                                          ⁢          1                )            
As mentioned above, for electrically small transmission lines the impedance transformation effect is commonly assumed to be negligible. This is consistent with assuming that for electrically small transmission lines the terms involving βΔz approach zero. The transmission line of FIG. 1 is electrically small when the length of the transmission line is much smaller than the wavelength of excitation of the power signal. When the terms involving βΔz are assumed to be zero the relationship between the input impedance Zin and the load impedance ZL reduces to:Zin≅ZL  (Equation 2)
The accuracy of Equation 2 depends strongly on the magnitude of ZL relative to Z0. For electrically small transmission lines tan(βΔz) is assumed to be approximately equal to βΔz as per the small angle approximation. The relationship between the input impedance Zin and the load impedance ZL can then be expressed as:
                              Z          in                ≅                              Z            0                    ⁢                                                    Z                L                            +                                                jZ                  0                                ⁢                                                                  ⁢                βΔ                ⁢                                                                  ⁢                z                                                                    Z                0                            +                                                jZ                  L                                ⁢                                                                  ⁢                βΔ                ⁢                                                                  ⁢                z                                                                        (                  Equation          ⁢                                          ⁢          3                )            
When |ZL|βΔz<<|Z0|, Equation 3 can be simplified to the following:Zin≅ZL+jZ0βΔz  (Equation 4)
In this scenario, the delay associated with the transmission line causes a shift in the wavelengths-toward-generator, or clockwise direction around the Smith chart. Since |ZL|<<|Z0|, the result is an effective series inductance.
When |ZL|>>|Z0βΔz, Equation 3 can be simplified to the following:
                              Z          in                ≅                              Z            L                                1            +                                          jZ                L                            ⁢                              Y                0                            ⁢              βΔ              ⁢                                                          ⁢              z                                                          (                  Equation          ⁢                                          ⁢          5                )            
In Equation 5, Y0 is Z0−1. In this scenario, the delay associated with the transmission line causes a shift in the wavelengths-toward-generator, or clockwise direction around the Smith chart. Since |ZL|>>|Z0|, the result is an effective parallel capacitance. In both Equations 4 and 5, the transmission line causes a phase delay in the circuit 8.
This phase delay may be problematic for RF systems in which the goal is to efficiently deliver power to a variable load impedance, such as wireless power transfer systems that transfer power via resonant coupling. In such a wireless power transfer system, a power source outputs a sinusoidal power signal that drives a transmit resonator which transfers power to a receive resonator. The system input impedance exhibits an inverse relationship with the distance between the transmit and receive resonators. For a distance of 10 cm to 30 cm, a system input impedance of 1100Ω to 20Ω may be observed.
Table 1 below is an example set of input impedance Zin values calculated for a given set of load impedance ZL values for a Δz=0.3048 m (12″) transmission line. In this example, the load impedance ZL values are selected from the set including 20Ω, 50Ω, 150Ω, 300Ω, 510Ω, 750Ω and 1100Ω. The characteristic impedance Z0 of the transmission line is 50Ω. The phase constant β of the transmission line is 0.426 rad/m. Assuming a propagation velocity of ⅔ of the speed of light, this phase constant β corresponds to an excitation frequency of 13.56 MHz. As previously stated, the physical length Δz of the transmission line is 0.3048 m (12″). The wavelength λ of the power signal from the power source is 14.6304 m (576″). Thus, the physical length Δz of the transmission line is λ/48. Given these values, Zin can be calculated according to Equation 1 for the various ZL values in the given range.
TABLE 1ZL (Ω)Zin (Ω)2020 + j6 5050 + j0 150132 − j45 300189 − j142510187 − j242750158 − j3021100121 − j341
As shown in Table 1, the presence of the transmission line has an impact on the input impedance Zin of the circuit 8 except for the matched impedance case in which the load impedance ZL is equal to the characteristic impedance Z0 of the transmission line. In the matched impedance case, both the load impedance ZL and the input impedance Zin are 50Ω. In the other cases (i.e. when the characteristic and load impedances are mismatched), the presence of the transmission line adds an additional reactance to the input impedance Zin. Thus, the load impedance ZL and the input impedance Zin are not equal. For example when the load impedance ZL is 300Ω, the input impedance Zin is 189−j142Ω.
In the context of antennas or resonators in wireless power transfer systems that transfer power via resonant coupling, the additional reactance is analogous to detuning the transmitter of the wireless power transfer system, resulting in less efficient power transfer. It is therefore an object of an aspect of the subject disclosure to at least partially mitigate the impedance transformation effect of transmission lines on electrical systems.